Linear Algebra

Topic: Input-Output Analysis 63
Topic: Input-Output Analysis
An economy is an immensely complicated network of interdependences. Changes
in one part can ripple out to affect other parts. Economists have struggled to
be able to describe, and to make predictions about, such a complicated object.
Mathematical models using systems of linear equations have emerged as a key
tool. One is Input-Output Analysis, pioneered by W. Leontief, who won the
1973 Nobel Prize in Economics.
Consider an economy with many parts, two of which are the steel industry
and the auto industry. As they work to meet the demand for their product from
other parts of the economy, that is, from users external to the steel and auto
sectors, these two interact tightly. For instance, should the external demand
for autos go up, that would lead to an increase in the auto industry’s usage of
steel. Or, should the external demand for steel fall, then it would lead to a fall
in steel’s purchase of autos. The type of Input-Output model we will consider
takes in the external demands and then predicts how the two interact to meet
those demands.
We start with a listing of production and consumption statistics. (These
numbers, giving dollar values in millions, are excerpted from [Leontief 1965],
describing the 1958 U.S. economy. Today’s statistics would be quite different,
both because of inflation and because of technical changes in the industries.)
value of
steel
value of
auto
used by
steel
used by
auto
used by
others total
5 395 2 664 25 448
48 9 030 30 346
For instance, the dollar value of steel used by the auto industry in this year is
2, 664 million. Note that industries may consume some of their own output.
We can fill in the blanks for the external demand. This year’s value of the
steel used by others this year is 17, 389 and this year’s value of the auto used
by others is 21, 268. With that, we have a complete description of the external
demands and of how auto and steel interact, this year, to meet them.
Now, imagine that the external demand for steel has recently been going up
by 200 per year and so we estimate that next year it will be 17, 589. Imagine
also that for similar reasons we estimate that next year’s external demand for
autos will be down 25 to 21, 243. We wish to predict next year’s total outputs.
That prediction isn’t as simple as adding 200 to this year’s steel total and
subtracting 25 from this year’s auto total. For one thing, a rise in steel will
cause that industry to have an increased demand for autos, which will mitigate,
to some extent, the loss in external demand for autos. On the other hand, the
drop in external demand for autos will cause the auto industry to use less steel,
and so lessen somewhat the upswing in steel’s business. In short, these two
industries form a system, and we need to predict the totals at which the system
as a whole will settle.